The DBAR problem, or the

\bar

-problem, is the problem of solving the

differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, a ...

\bar f (z, \bar) = g(z)

for the function

f(z,\bar)

, where

g(z)

is assumed to be known and

z = x + iy

is a complex number in a

domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined ** Domain of definition of a partial function **Natural domain of a partial function **Domain of holomorphy of a function *Do ...

R\subseteq \Complex

. The operator

\bar

is called the DBAR operator

\bar = \frac \left(\frac + i \frac \right)

The DBAR operator is nothing other than the complex conjugate of the operator

\partial=\frac = \frac \left(\frac - i \frac \right)

denoting the usual differentiation in the complex

z

-plane. The DBAR problem is of key importance in the theory of

integrable systems In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first i ...

and generalizes the

Riemann–Hilbert problem In mathematics, Riemann–Hilbert problems, named after Bernhard Riemann and David Hilbert, are a class of problems that arise in the study of differential equations in the complex plane. Several existence theorems for Riemann–Hilbert problem ...

References

{{Reflist DBAR problem